To determine which probability is approximately equal to 0.2957 using the given standard normal table, we start by identifying the probability values associated with the z-scores given in each answer choice.
Given the table:
\begin{tabular}{|c|c|}
\hline[tex]$z$[/tex] & Probability \\
\hline 0.00 & 0.5000 \\
\hline 0.25 & 0.5987 \\
\hline 0.50 & 0.6915 \\
\hline 0.75 & 0.7734 \\
\hline 1.00 & 0.8413 \\
\hline 1.25 & 0.8944 \\
\hline 1.50 & 0.9332 \\
\hline 1.75 & 0.9599 \\
\hline \hline
\end{tabular}
We are asked to evaluate the following probabilities:
1. [tex]\(P(-1.25 \leq z \leq 0.25)\)[/tex]
2. [tex]\(P(-1.25 \leq z \leq 0.75)\)[/tex]
3. [tex]\(P(0.25 \leq z \leq 1.25)\)[/tex]
4. [tex]\(P(\cap 75
Let's determine each probability:
1. [tex]\(P(-1.25 \leq z \leq 0.25)\)[/tex]
- From the standard normal table, the probability for [tex]\(z = -1.25\)[/tex] (by symmetry of the normal distribution) will be [tex]\(1 - 0.8944 = 0.1056\)[/tex].
- Therefore, [tex]\(P(-1.25 \leq z \leq 0.25) = P(z \leq 0.25) - P(z \leq -1.25) = 0.5987 - 0.1056 = 0.4931\)[/tex].
2. [tex]\(P(-1.25 \leq z \leq 0.75)\)[/tex]
- From the standard normal table, the probability for [tex]\(z = 0.75\)[/tex] is 0.7734.
- Therefore, [tex]\(P(-1.25 \leq z \leq 0.75) = P(z \leq 0.75) - P(z \leq -1.25) = 0.7734 - 0.1056 = 0.6678\)[/tex].
3. [tex]\(P(0.25 \leq z \leq 1.25)\)[/tex]
- From the standard normal table, the probability for [tex]\(z = 1.25\)[/tex] is 0.8944.
- Therefore, [tex]\(P(0.25 \leq z \leq 1.25) = P(z \leq 1.25) - P(z \leq 0.25) = 0.8944 - 0.5987 = 0.2957\)[/tex].
4. [tex]\(P(\cap 75
- Due to the unclear representation, this option is not considered valid.
By examining the calculations, the probability [tex]\(P(0.25 \leq z \leq 1.25)\)[/tex] results in approximately 0.2957. Therefore, the correct answer is:
[tex]\[ \boxed{P(0.25 \leq z \leq 1.25)} \][/tex]
